NOTES ON THE THEORY OF THE TRANSFORMER 277 



In this case it is much more difficult to express the hysteresis empir- 

 ically. In most cases the first term in the value of y is the largest. A 

 term of the same nature as before will, in this case, suffice to express 

 the hysteresis approximated. We can then write for the total flux of 

 magnetic induction 



p = A cos (U + ei) + Buy + Cn 3 y* + Dtfy 5 + etc. 



Problem 1. To find the electromotive force necessary to make the 

 electric current a sine curve in a transformer without secondary. Let 

 the resistance be E, and make y = c sin (bt). Then Maxwell's equation 

 becomes 



*= 



Substituting the value of y we have 



E= (RcAbn} sin (bt} + Bncb cos (bt} + 3 Cn 3 sin 2 (bt) cos bt + etc. 

 But 



Sin *bt cos bt = \ (cos bt cos 3 U} 

 Sin *bt cos U = jig. (cos 5 bt 3 cos 3 U + 2 cos bt) 

 Si n 6 ^ cos bt = etc. 



Hence the electromotive force that must be given to the circuit must 

 contain not only the given frequency of the current but also frequencies 

 of 3, 5, 7, etc., times as many. In other words, the odd harmonics. 



Problem 2. Transformer without secondary, the electromotive force 

 being a sine curve. 



E sin U = Ry + n . 

 ct t 



First it is to be noted that when we place in this equation the general 

 value of y and make the coefficients of like functions of bt zero, all the 

 even harmonics will strike out. 



Hence the value of the electric current will be 



y = a 1 sin(W + i) + 3 sin (3 bt + e 3 )+a 6 sin (5bt + e t )+. 

 Substituting this value in the value for p, the equation is theoretically 

 sufficient to determine a v a z , etc., and e lf e 3) etc. The equations are 

 cubic or of higher order and the solution can only be approximate and I 

 have not thought it worth while to go further with the calculation. 

 However, it is easy to draw the following conclusion: 



1. A simple harmonic current through an iron transformer will pro- 

 duce a secondary electromotive force and current, or both, which con- 

 tain not only the fundamental period but the higher odd harmonics. 



